Dr. Niklas Kolbe

Academic positions

• PostDoc, IGPM, RWTH Aachen University (Nov 2021 - now)
  in the group of Prof. Dr. Michael Herty
• JSPS PostDoc Fellow, Kanazawa University (Jul 2019 - Oct 2021)
  in the group of Prof. Hirofumi Notsu
• PostDoc, Johannes Gutenberg University Mainz (Jan 2018- Jun 2019)
  in the group of Prof. Dr. Mária Lukácová-Medvidová
• PhD in Mathematics, Institute of Mathematics, Johannes Gutenberg University Mainz (Dec 2017)
  supervisors: Prof. Dr. Mária Lukácová-Medvidová, Prof. Dr. Nadja Hellmann, Dr. Nikolaos Sfakianakis

Research interests

• Finite volume methods for parabolic and hyperbolic PDEs
• Adaptive mesh movement/refinement
• Lagrangian and mass transport schemes
• Modeling of cancer invasion and protein pathways
• Networks of hyperbolic conservation laws

Preprints

[BK25]

A. Beckers and N. Kolbe. The Lax-Friedrichs method in one-dimensional hemodynamics, 2025. doi: 10.48550/arXiv.2501.16115.

[HKN24]

M. Herty, N. Kolbe and M. Neidlin. A one-dimensional model for aspiration therapy in blood vessels, 2024. doi: 10.48550/arXiv.2403.05494.

[KM24]

N. Kolbe and S. Müller. A relaxation approach to the coupling of a two-phase fluid with a linear-elastic solid, 2024. doi: 10.48550/arXiv.2409.05473.

Journal publications

[GK24]

J. Giesselmann and N. Kolbe. A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller–Segel model. IMA J. Numer. Anal.:drae073, 2024. doi: 10.1093/imanum/drae073.

[HK24]

M. Herty and N. Kolbe. Data-driven models for traffic flow at junctions. Math. Methods Appl. Sci., 47(11):8946–8968, 2024. doi: 10.1002/mma.10053.

[KHM24]

N. Kolbe, M. Herty and S. Müller. Numerical schemes for coupled systems of nonconservative hyperbolic equations. SIAM J. Numer. Anal., 62(5):2143–2171, 2024. doi: 10.1137/23M1615176.

[PMK⁺24]

K. S. Putri, T. Mizuochi, N. Kolbe and H. Notsu. Error estimates for first- and second-order lagrange–galerkin moving mesh schemes for the one-dimensional convection–diffusion equation. J. Sci. Comput., 101(2):37, 2024. doi: 10.1007/s10915-024-02673-4.

[HKM23a]

M. Herty, N. Kolbe and S. Müller. A central scheme for two coupled hyperbolic systems. Commun. Appl. Math. Comp., 2023. doi: 10.1007/s42967-023-00306-5.

[HKM23b]

M. Herty, N. Kolbe and S. Müller. Central schemes for networked scalar conservation laws. Netw. Heterog. Media, 18(1):310–340, 2023. doi: 10.3934/nhm.2023012.

[DKS⁺22]

A. Dietrich, N. Kolbe, N. Sfakianakis and C. Surulescu. Multiscale modeling of glioma invasion: from receptor binding to flux-limited macroscopic PDEs. Multiscale Model. Simul., 20(2):685–713, 2022. doi: 10.1137/21M1412104.

[FKN⁺22]

K. Futai, N. Kolbe, H. Notsu and T. Suzuki. A mass-preserving two-step Lagrange–Galerkin scheme for convection-diffusion problems. J. Sci. Comput., 92(2):37, 2022. doi: 10.1007/s10915-022-01885-w.

[KHB⁺22]

N. Kolbe, L. Hexemer, L.-M. Bammert, A. Loewer, M. Lukáčová-Medvid’ová and S. Legewie. Data-based stochastic modeling reveals sources of activity bursts in single-cell TGF-β signaling. PLoS Comput. Biol., 18(6):1–29, 2022. doi: 10.1371/journal.pcbi.1010266.

[KS22]

N. Kolbe and N. Sfakianakis. An adaptive rectangular mesh administration and refinement technique with application in cancer invasion models. J. Comput. Appl. Math., 416:114442, 2022. doi: 10.1016/j.cam.2022.114442.

[KSS⁺21]

N. Kolbe, N. Sfakianakis, C. Stinner, C. Surulescu and J. Lenz. Modeling multiple taxis: tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence. Discrete Contin. Dyn. Syst. Ser. B, 26(1):443–481, 2021. doi: 10.3934/dcdsb.2020284.

[CKL19]

J. A. Carrillo, N. Kolbe and M. Lukáčová-Medvid’ová. A hybrid mass transport finite element method for Keller-Segel type systems. J. Sci. Comput., 80(3):1777–1804, 2019. doi: 10.1007/s10915-019-00997-0.

[GKL⁺18]

J. Giesselmann, N. Kolbe, M. Lukáčová-Medvid’ová and N. Sfakianakis. Existence and uniqueness of global classical solutions to a two species cancer invasion haptotaxis model. Discrete. Cont. Dyn.-B, 23(10):4397–4431, 2018. doi: 10.3934/dcdsb.2018169.

[Kol18]

N. Kolbe. A Tumor Invasion Model for Heterogeneous Cancer Cell Populations: Mathematical Analysis and Numerical Methods. PhD thesis, Johannes Gutenberg University Mainz, 2018. doi: 10.25358/openscience-2836.

[SKH⁺17]

N. Sfakianakis, N. Kolbe, N. Hellmann and M. Lukáčová-Medvid’ová. A multiscale approach to the migration of cancer stem cells: mathematical modelling and simulations. Bull. Math. Biol., 79(1):209–235, 2017. doi: 10.1007/s11538-016-0233-6.

[HKS16]

N. Hellmann, N. Kolbe and N. Sfakianakis. A mathematical insight in the epithelial-mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix. Bull. Braz. Math. Soc., New Series, 47(1):397–412, 2016. doi: 10.1007/s00574-016-0147-9.

[KKS⁺16]

N. Kolbe, J. Kat’uchová, N. Sfakianakis, N. Hellmann and M. Lukáčová-Medvid’ová. A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion: The urokinase model. Appl. Math. Comput., 273:353–376, 2016. doi: 10.1016/j.amc.2015.08.023.

Book chapters and conference proceedings

[Kol24]

N. Kolbe. Influx ratio preserving coupling conditions for the networked Lighthill-Whitham-Richards model, 2024. doi: 10.48550/arXiv.2405.21005. Accepted in Proc. Appl. Math. Mech.

[KBK⁺24]

N. Kolbe, M. Berghaus, E. Kalló, M. Herty and M. Oeser. A microscopic on-ramp model based on macroscopic network flows. Appl. Sc., 14(19):9111, 2024. doi: 10.3390/app14199111.

[Kol23]

N. Kolbe. Numerical relaxation limit and outgoing edges in a central scheme for networked conservation laws. Proc. Appl. Math. Mech., 23(1):e202200150, 2023. doi: 10.1002/pamm.202200150.

[BKS18]

A. Brunk, N. Kolbe and N. Sfakianakis. Chemotaxis and Haptotaxis on Cellular Level. In C. Klingenberg and M. Westdickenberg, editors, Theory, Numerics and Applications of Hyperbolic Problems I. Volume 236, pages 249–261. Springer International Publishing, Cham, 2018. doi: 10.1007/978-3-319-91545-6˙20.

[KLS⁺17]

N. Kolbe, M. Lukáčová-Medvid’ová, N. Sfakianakis and B. Wiebe. Numerical Simulation of a Contractivity Based Multiscale Cancer Invasion Model. In A. Gerisch, R. Penta and J. Lang, editors, Multiscale Models in Mechano and Tumor Biology. Volume 122, pages 73–91. Springer International Publishing, 2017. doi: 10.1007/978-3-319-73371-5˙4.

More information including full text links to my publications can be found on my other homepage.